I am sorry to hear that your puppy Buster is not eating well and has vomited several times today, including a treat.
He is likely dehydrated because he has been vomiting, but if even water is making him vomit you need to take it away from him for now.
In most cases vomiting is triggered by eating something they should not, too much table food, too many treats or something they find outdoors.
More serious causes of vomiting at this age include toxin ingestions, viral or bacterial infections, a dietary allergy or sensitivity, inflammatory bowel disease, congenital internal organ failure (kidney or liver disease), or a full or partial gastrointestinal obstruction. If he were a middle aged to older fellow then an abdominal mass would also be a possible cause, but is much less likely at his young age.
In a young dog, especially if he never finished his vaccine series, a viral infection such as Parvo virus, toxin ingestion, or a foreign body leading to a partial or full gastrointestinal obstruction would be the most likely cause. Parvo virus does eventually cause bloody diarrhea, but generally we see vomiting first, and increased thirst due to dehydration. And there are other gastrointestinal viruses besides Parvo or bacterial infections that can cause vomiting too.
Because he is vomiting repeatedly ideally he would see a veterinarian now. At this stage he may need intensive care (intravenous fluids, injectable medications) that will be difficult to properly manage at home. I think he needs an injection to ease his nausea and may benefit from injectable antibiotics and fluids too.
If that isn't possible for whatever reason there are some things we can try at home, but if he's not improving quickly he should see a veterinarian for an examination, some diagnostic testing, intravenous fluids and injectable medication to settle his stomach.
To try and settle his stomach you can give either:
1) Pepcid ac (famotidine) at a dose of one half of a 10mg tablet per 10 to 20 pounds of body weight every 12 hours
2) Prilosec (omeprazole) at a dose of one quarter of a 20mg tablet per 10 to 20 pounds of body weight every 24 hours
These are acid reducers and may help settle his stomach and get him feeling better and hopefully get his appetite back. Either one can be given for several days if necessary.
I would pick up all food for now and water for a couple hours to allow his stomach to settle after the acid reducers.
In a couple hours when you give him water make sure it is in small amounts only. If he drinks too much too quickly that can lead to vomiting. You can also offer ice cubes to lick. To get some electrolytes in you can offer him a 50:50 mix of pedialyte and water.
If there is no vomiting for 12 hours offer a bland diet of 1/3 boiled, minced, white skinless chicken or boiled, lean hamburger and 2/3 boiled, white rice mixed with some low salt chicken or beef broth to make it easy to lap up and swallow. I recommend adding probiotics like Fortiflora, Proviable or Benebac to his meals daily to re-establish normal gut bacteria levels too.
If he refuses the bland diet mix you can offer a little meat baby food. If he refuses both then he needs veterinary care.
But if things go well and he does eat and doesn't vomit feed him the bland diet for 2 to 3 days then slowly start to mix back in his regular food, a little more at each meal. It should take about 5 to 7 days to slowly convert him back to his regular diet.
If he continues to vomit even with the acid reducers, runs a fever (more than 103F rectally), or has a lower then normal temperature (less then 99F), has a tense painful belly, has bloody diarrhea, or if he refuses to eat he should see a veterinarian for an examination, diagnostics, injectable anti-nausea drugs intravenous fluids and supportive care.
Please let me know if you have any further questions.